^†^†thanks: This research was supported by by NFSC, grant 12201381.

A relative Nadel-type vanishing theorem

Jingcao Wu [email protected] School of Mathematica, Shanghai University of Finance and Economics, Shanghai 200433, People’s Republic of China

Abstract

In this paper we prove a relative Nadel-type vanishing theorem on Kähler morphisms. Then we discuss its applications on harmonic bundles. In particular, it gives a relative vanishing concerning Saito’s S-sheaves.

:

32J25 (primary), 32L20 (secondary).

keywords:

vanishing theorem, invariance of plurigenera, asymptotic multiplier ideal sheaf.

1 Introduction

The Nadel vanishing theorem is a powerful tool in complex geometry. It says that

Theorem 1.1 ((c.f. [Nad90, Dem93])).

Let $X$ be a projective manifold of dimension $n$ , and let $L$ be a holomorphic line bundle on $X$ . Fix a Kähler metric $\omega$ on $X$ . Assume that $L$ is equipped with a singular Hermitian metric $\varphi$ with $i\Theta_{L,\varphi}\geqslant\varepsilon\omega$ for some $\varepsilon>0$ . Then

H^{q}(X,K_{X}\otimes L\otimes\mathscr{I}(\varphi))=0

for $q>0$ .

Here $\mathscr{I}(\varphi)$ refers to the multiplier ideal sheaf [Nad90] associated to $\varphi$ . Note that in Theorem 1.1, $L$ is by definition a big line bundle [Dem12]. So we will also directly say that $(L,\varphi)$ is a big line bundle.

However, one of the limitations of Theorem 1.1 is that $L$ should be big. It is then asked to generalise it. One direction is to consider the line bundle $L$ possessing weaker positivity, which is fully studied by [Cao14, Eno93, Mat14, Mat15a, Mat15b, Mat18, Wu17, Wu22]. Another direction should be the relative case. See [Laz04b], Generalisations 9.1.22 and 11.2.15, for example. In this paper, we will provide a relative version of the Nadel-type vanishing theorem. Before stating the main result, we should fix some notations and conventions first.

Throughout this paper, unless otherwise stated, $f:X\rightarrow Y$ is a proper, locally Kähler morphism from a complex manifold $X$ to a reduced, irreducible analytic space $Y$ . (Remember that $f$ is locally Kähler if $f^{-1}(U)$ is a Kähler space for any relatively compact, open subset $U$ of $Y$ .) Every connected component of $X$ is mapped surjectively to $Y$ . $X^{o}$ is a Zariski open subset of $X$ , and $(E,h)$ is a tame, Nakano semi-positive vector bundle over $X^{o}$ . Let $i:X^{o}\rightarrow X$ be the natural embedding, and define

E(h)_{x}:=\{s\in(i_{\ast}E)_{x};|s|_{h}\textrm{ is locally }L^{2}\textrm{-integrable against the Lebesgue measure}\}.

It is a coherent sheaf on $X$ by [ShZ21a], Proposition 2.9. In the end, we denote by $l$ the dimension of a general fibre of $f$ . Then the main theorem is as follows.

Theorem 1.2.

Let $L$ be a holomorphic line bundle on $X$ with $\kappa(L,f)\geqslant 0$ . Assume that there exists a section $s$ of some multiple $L^{m}$ such that $\{s=0\}\subseteq X^{o}$ . Then

R^{q}f_{\ast}(K_{X}\otimes L\otimes E(h)\otimes\mathscr{I}(f,\|L\|))=0

for $q>l-\kappa(L,f)$ .

Here $\mathscr{I}(f,\|L\|)$ is the relative asymptotic multiplier ideal sheaf and $\kappa(L,f)$ is the relative Iitaka dimension of $L$ . Note that when $X^{o}=X$ , $E$ is always tame and $E(h)=E$ . At this time our result extends Generlisation 11.2.15 in [Laz04b].

Corollary 1.1.

Let $f:X\rightarrow Y$ be a proper, locally Kähler morphism from a complex manifold $X$ to a reduced, irreducible analytic space $Y$ . Every connected component of $X$ is mapped surjectively to $Y$ . Let $E$ be a Nakano semi-positive vector bundle over $X$ . Let $L$ be a holomorphic line bundle on $X$ with $\kappa(L,f)\geqslant 0$ . Then

R^{q}f_{\ast}(K_{X}\otimes L\otimes E\otimes\mathscr{I}(f,\|L\|))=0

for $q>l-\kappa(L,f)$ .

In general, as is explained in [ScY20], $E$ can be interpreted as a vector bundle over $X$ equipped with a singular Hermitian metric $h$ , and $E(h)$ is a higher rank analogy of a singular Hermitian line bundle tensoring with its associated multiplier ideal sheaf. Moreover, the reflexivity of $X^{o}$ allows fruitful applications, and the one on harmonic bundles is extremely interesting among them.

Remember that the harmonic bundles are important objects in non-abelian Hodge theory. More precisely, C. Simpson [Sim92] used it to establish a correspondence between local systems and semistable Higgs bundles with vanishing Chern classes. Then our vanishing implies

Corollary 1.2.

Let $L$ be a holomorphic line bundle with $\kappa(L,f)\geqslant 0$ . Assume that there exists a section $s$ of some multiple $L^{m}$ such that $\{s=0\}\subseteq X^{o}$ . Let $(H,\theta,h)$ be a tame harmonic bundle over $X^{o}$ , and let $E$ be a subbundle of $H$ with vanishing second fundamental form and $\bar{\theta}(E)=0$ . Then

R^{q}f_{\ast}(K_{X}\otimes L\otimes E(h)\otimes\mathscr{I}(f,\|L\|))=0

for $q>l-\kappa(L,f)$ .

The $S$ -sheaf is a typical example that satisfies the conditions in Corollary 1.2. Recall that M. Saito constructed a coherent sheaf $S(\mathrm{IC}_{X}(\mathbb{V}))$ in [Sai91], which is called $S$ -sheaf, for a real variation of polarized Hodge structure $(\mathbb{V},\nabla,\mathcal{F}^{\cdot},S)$ [CKS86] on $X^{o}$ . It plays a key role to solve Kollar’s conjecture [Ko86b]. For this sheaf, we have

Corollary 1.3.

Let $L$ be a holomorphic line bundle on $X$ with $\kappa(L,f)\geqslant 0$ . Assume that there exists a section $s$ of some multiple $L^{m}$ such that $\{s=0\}\subseteq X^{o}$ . Then

R^{q}f_{\ast}(S(\mathrm{IC}_{X}(\mathbb{V}))\otimes L\otimes\mathscr{I}(f,\|L\|))=0

for $q>l-\kappa(L,f)$ .

It can also be viewed as a refinement of the torsion-freeness part in Kollar’s conjecture, and a generalisation of [ShZ21b], Theorem 1.2.

Now we outline the strategy to prove Theorem 1.2. Firstly, we prove the following Kollár-type injectivity and torsion-free theorem.

Theorem 1.3.

Let $L$ be a holomorphic line bundle on $X$ with $\kappa(L,f)\geqslant 0$ . For a (non-zero) section $s$ of some multiple $L^{m-1}$ such that $\{s=0\}\subseteq X^{o}$ , the multiplication map induced by the tensor product with $s$

\Phi:R^{q}f_{\ast}(K_{X}\otimes L\otimes E(h)\otimes\mathscr{I}(f,\|L\|))\rightarrow R^{q}f_{\ast}(K_{X}\otimes L^{m}\otimes E(h)\otimes\mathscr{I}(f,\|L^{m}\|))

is well-defined and injective for any $q\geqslant 0$ . In particular, $R^{q}f_{\ast}(K_{X}\otimes L\otimes E(h)\otimes\mathscr{I}(f,\|L\|))$ is torsion-free for every $q$ .

The injectivity theorem has been fully studied in [Eno93, Fuj12, Ko86a, Ko86b, Mat14, Mat15a, Mat15b, Mat18, ShZ21a]. However, we cannot directly apply any result among these papers to obtain Theorem 1.3. The reason is that in general there does not exist a metric $\varphi$ on $L$ such that

i\Theta_{L,\varphi}\geqslant 0\quad\textrm{and}\quad\mathscr{I}(\varphi)=\mathscr{I}(f,\|L\|).

We will follow the idea of [Tak95] to develop a harmonic theory concerning the singular Hermitian metric, and use it to finish the proof.

Note that the injectivity and torsion-free theorem (i.e. Theorem 1.3) together with the Kollár-type vanishing theorem and decomposition theorem is usually called Kollár’s package. In particular, the injectivity and torsion-free theorem is the key among them. In other words,

K_{X}\otimes L\otimes E(h)\otimes\mathscr{I}(f,\|L\|))

should moreover satisfy Kollár’s package. We will leave this work in the future.

Return to our main result, Theorem 1.2 is now a combination of Theorem 1.3 and the Nadel-type vanishing theorem (Theorem 1.3) in [Wu22].

This paper is organised as follows. We first recall some background materials, including the asymptotic multiplier ideal sheaf, tame vector bundles and so on. Then, we proceed to develop the harmonic theory associated with singular Hermitian metrics in Sect.3. Based on this theory, we will prove Theorems 1.2 and 1.3 in Sect.4. In the end, we discuss the applications.

Finally, I would like to mention a series of brilliant work concerning Kollár’s package due to Junchao Shentu and Chen Zhao, including [ShZ21a, ShZ21b, ShZ22]. After a very early prototype of Theorem 1.2 was finished, the author fortunately learned these papers, which inspired the further study of the applications of our vanishing theorem. So I own a lot to them.

2 Preliminary

In this section we will introduce some basic materials.

2.1 The asymptotic multiplier ideal sheaf

This part is mostly collected from [Laz04b].

First recall the definition of the multiplier ideal sheaf associated to an ideal sheaf $\mathfrak{a}\subset\mathcal{O}_{X}$ and a positive real number $c$ . Let $\mu:\tilde{X}\rightarrow X$ be a smooth modification such that $\mu^{\ast}\mathfrak{a}=\mathcal{O}_{\tilde{X}}(-E)$ , where $E$ has the simple normal crossing support. Then the multiplier ideal sheaf is defined as

\mathscr{I}(c\cdot\mathfrak{a}):=\mu_{\ast}\mathcal{O}_{\tilde{X}}(K_{\tilde{X}/X}-\lfloor cE\rfloor).

Here $K_{\tilde{X}/X}$ is the relative canonical bundle and $\lfloor E\rfloor$ means the round-down.

Now let $f:X\rightarrow Y$ be a proper, locally Kähler and surjective morphism from a complex manifold $X$ to a reduced, irreducible analyitc space $Y$ . Suppose that $L$ is a line bundle on $X$ whose restriction to a general fibre of $f$ has non-negative Iitaka dimension. For a positive integer $k$ , there is a naturally defined homomorphism

\rho_{k}:f^{\ast}f_{\ast}(L^{k})\rightarrow L^{k}.

The relative base-ideal $\mathfrak{a}_{k,f}$ of $|L^{k}|$ is then defined as the image of the induced homomorphism

f^{\ast}f_{\ast}L^{k}\otimes L^{-k}\rightarrow\mathcal{O}_{X}.

Hence for a given positive real number $c$ , we have the multiplier ideal sheaf $\mathscr{I}(\frac{c}{k}\cdot\mathfrak{a}_{k,f})$ which is also denoted by $\mathscr{I}(f,\frac{c}{k}|L^{k}|)$ . It is not hard to verify that for every integer $p\geqslant 1$ one has the inclusion

\mathscr{I}(f,\frac{c}{k}|L^{k}|)\subseteq\mathscr{I}(f,\frac{c}{pk}|L^{pk}|).

Therefore the family of ideals

\{\mathscr{I}(f,\frac{c}{k}|L^{k}|)\}_{(k\geqslant 0)}

has a unique maximal element from the ascending chain condition on ideals.

Definition 2.1.

The relative asymptotic multiplier ideal sheaf associated to $f$ , $c$ and $|L|$ ,

\mathscr{I}(f,c\|L\|)

is defined to be the unique maximal member among the family of ideals $\{\mathscr{I}(f,\frac{c}{k}|L^{k}|)\}$ .

Next, we explain the analytic counterpart of the relative multiple ideal sheaf. By definition,

\mathscr{I}\bigl{(}f,c\|L\|\bigr{)}=\mathscr{I}\bigl{(}f,\frac{c}{k}|L^{k}|\bigr{)}=\mathscr{I}\bigl{(}\frac{c}{k}\cdot\mathfrak{a}_{k,f}\bigr{)}

for some $k$ . In this case, we will say that $k$ computes $\mathscr{I}(f,c\|L\|)$ . Let $U$ be a Stein open subset of $Y$ . By definition, we can pick $\{u_{1},...,u_{m}\}$ in $\Gamma(f^{-1}(U),L^{k})$ which generate $\mathfrak{a}_{k,f}$ on $f^{-1}(U)$ . Let $\varphi_{U}=\frac{1}{k}\log(|u_{1}|^{2}+\cdots+|u_{m}|^{2})$ which is a singular metric on $L|_{f^{-1}(U)}$ . We verify that

\mathscr{I}(f,\frac{c}{k}|L^{k}|)=\mathscr{I}(c\varphi_{U})\textrm{ on }f^{-1}(U).

Indeed, let $\mu:\tilde{X}\rightarrow X$ be a smooth modification of $\mathfrak{a}_{k,f}$ . Then $\mu^{\ast}\mathfrak{a}_{k,f}=\mathcal{O}_{\tilde{X}}(-E)$ such that $E+\textrm{except}(\mu)$ has the simple normal crossing support. Here $\textrm{except}(\mu)$ is the exceptional divisor of $\mu$ . Now it is computed in [Dem12] that

\mathscr{I}\bigl{(}c\varphi_{U}\bigr{)}=\mu_{\ast}\mathcal{O}_{\tilde{X}}\bigl{(}K_{\tilde{X}/X}-\lfloor\frac{c}{k}E\rfloor\bigr{)}\textrm{ on }f^{-1}(U)

which coincides with the definition of $\mathscr{I}(f,\frac{c}{k}|L^{k}|)$ . Furthermore, if $v_{1},...,v_{m}$ are alternative generators and $\psi_{U}=\frac{1}{k}\log(|v_{1}|^{2}+\cdots+|v_{m}|^{2})$ , obviously we have $\mathscr{I}(c\varphi_{U})=\mathscr{I}(c\psi_{U})$ . Hence all the $\mathscr{I}(c\varphi_{U})$ patch together to give a globally defined multiplier ideal sheaf $\mathscr{I}(c\varphi)$ such that

\mathscr{I}(c\varphi)=\mathscr{I}(f,\frac{c}{k}|L^{k}|)=\mathscr{I}(f,c\|L\|).

Note that $\{f^{-1}(U),\varphi_{U}\}$ does not give a globally defined metric on $L$ in general. The $\varphi$ is interpreted as the collection of functions $\{f^{-1}(U),\varphi_{U}\}$ by abusing the notation, which is called the collection of (local) singular metrics on $L$ associated to $\mathscr{I}(f,c\|L\|)$ . Certainly it depends on the choice of $k$ and is not unique.

The following elementary property is due to [Laz04b].

Proposition 2.1.

Let $L_{1},L_{2}$ be holomorphic line bundles on $X$ with $\kappa(L_{1},f),\kappa(L_{2},f)\geqslant 0$ . Let $m$ and $k$ be non-negative integers. Let $\mathfrak{a}_{m,f}$ be the base-ideal of $|L^{m}_{1}|$ relative to $f$ . Then

\mathfrak{a}_{m,f}\cdot\mathscr{I}(f,\|L^{k}_{2}\|)\subseteq\mathscr{I}(f,\|L^{m}_{1}\otimes L^{k}_{2}\|).

2.2 Tame bundle

Let $X^{o}$ be a Zariski open subset of $X$ , and let $(E,h)$ be a Hermitian vector bundle over $X^{o}$ .

Definition 2.2.

$(E,h)$ is tame on $X$ , if for every $x\in X$ there exist an open neighbourhood $U$ containing $x$ , a proper bimeromorphic morphism $\pi:\tilde{U}\rightarrow U$ which is biholomorphic on $U\cap X^{o}$ , and a Hermitian vector bundle $(Q,h_{Q})$ on $\tilde{U}$ such that

(1)

$\pi^{\ast}E|_{U\cap X^{o}}\subseteq Q|_{\pi^{-1}(U\cap X^{o})}$ ;
(2)

There is a Hermitian metric $h^{\prime}_{Q}$ on $Q|_{\pi^{-1}(U\cap X^{o})}$ with $C_{1}\pi^{\ast}h\leqslant h^{\prime}_{Q}|_{\pi^{\ast}E}\leqslant C_{2}\pi^{\ast}h$ and

$(\sum^{r}_{i=1}|\pi^{\ast}f_{i}|^{2})^{c}h_{Q}\leqslant Ch^{\prime}_{Q}$

for some $c,C,C_{1},C_{2}\in\mathbb{R}_{+}$ . Here $f_{1},...,f_{r}$ are arbitrary defining functions of $U\setminus X^{o}$ .

In this paper, tame bundles are constructed from harmonic bundles. So we should also recall this notion. Let $(H,D)$ be a holomorphic vector bundle on $X$ with a flat connection $D$ . Let $h_{H}$ be an arbitrary Hermitian metric on $H$ .

Decompose $D=D^{1,0}+D^{0,1}$ into operators of type $(1,0)$ and $(0,1)$ respectively. Let $\delta^{\prime}_{H}$ and $\delta^{\prime\prime}_{H}$ be the unique operators of type $(1,0)$ and $(0,1)$ such that the connections $D^{1,0}+\delta^{\prime}_{H}$ and $D^{0,1}+\delta^{\prime\prime}_{H}$ preserve $h_{H}$ . Denote $\theta=\frac{1}{2}(D^{1,0}-\delta^{\prime}_{H})$ , $D^{c}_{H}=\delta^{\prime\prime}_{H}-\delta^{\prime}_{H}$ and $\Theta_{H}(D)=DD^{c}_{H}+D^{c}_{H}D$ .

Remark 2.1.

$\Theta_{H}(D)$ is called the pseudo-curvature associated with $h_{H}$ . In this paper, we will denote the curvature associated with $h_{H}$ by $\Theta_{H,h_{H}}$ to distinguish these two notions.

Definition 2.3.

$(H,\theta,h_{H})$ is called a harmonic bundle if $\Theta_{H}(D)=0$ . In this case, $h_{H}$ is called a harmonic metric.

This notion plays an important role in non-abelian Hodge theory. More precisely, it helps to establish the correspondence between local systems and semistable Higgs bundles with vanishing Chern classes.

Theorem 2.1 ((c.f. [Sim92])).

There is an equivalence of categories between the category of semisimple flat bundles on $X$ and the category of polystable Higgs bundles with $c_{1}(E)=0$ and $c_{2}(E)=0$ , both being equivalent to the category of harmonic bundles

Now we are ready to reformulate two canonical types of tame bundles. This part should be well-known to experts, and we recommend [ShZ21a] for an explicit exposition.

Proposition 2.2.

Let $(H,\theta,h_{H})$ be a tame harmonic bundle over $X^{o}$ . If $E$ is a holomorphic subbundle with vanishing second fundamental form and $\bar{\theta}(E)=0$ , then $(E,h_{H}|_{E})$ is a tame hermitian vector bundle with Nakano semi-positive curvature.

In particular, we have

Proposition 2.3.

A real variation of polarized Hodge structure $(\mathbb{V},\nabla,\mathcal{F}^{\cdot},S)$ on $X^{o}$ is a tame harmonic bundle.

3 The harmonic theory

In this section, we develop the harmonic theory concerning the singular Hermitian metrics in order to prove Theorem 1.3. It is mainly inspired by [Tak95]. We prefer to first consider the case that $E=\mathcal{O}_{X^{o}}$ , in order to make it easy to understand. However, the whole things should be valid after tensoring with a non-trivial $E(h)$ as is indicated in [Tak95].

3.1 Background

Let $(X,\omega)$ be a Kähler manifold of dimension $n$ , and let $L$ be a holomorphic line bundle on $X$ endowed with a smooth Hermitian metric $\varphi$ . The pointwise inner product $\langle\cdot,\cdot\rangle_{\varphi,\omega}$ on $A^{p,q}(X,L)$ is defined by the equation:

\langle\alpha,\beta\rangle_{\varphi,\omega}dV_{\omega}:=\alpha\wedge\ast\bar{\beta}e^{-\varphi}

for $\alpha,\beta\in A^{p,q}(X,L)$ . The $L^{2}$ -inner product is defined by

(\alpha,\beta)_{\varphi,\omega}:=\int_{X}\langle\alpha,\beta\rangle_{\varphi,\omega}dV_{\omega}

for $\alpha,\beta\in A^{p,q}(X,L)$ , and the norm $\|\cdot\|_{\varphi,\omega}$ is induced by $(\cdot,\cdot)_{\varphi,\omega}$ . The standard operators such as $\bar{\partial}$ , $\partial_{\varphi}$ , $\bar{\partial}^{\ast}_{\varphi}:=-\ast\partial_{\varphi}\ast$ as well as $L$ , $\Lambda$ , etc., in Kähler geometry are defined with respect to $(\cdot,\cdot)_{\varphi,\omega}$ on $X$ . In particular, for a smooth $(s,t)$ -form $\gamma$ , let $e(\gamma)$ be the morphism

\gamma\wedge\cdot:A^{p,q}(X,L)\rightarrow A^{p+s,q+t}(X,L).

We then define the operator on $A^{p,q}(X,L)$ by $e(\gamma)^{\ast}:=(-1)^{(p+q)(s+t+1)}\ast e(\bar{\gamma})\ast$ . Obviously, $e(\gamma)^{\ast}$ is the adjoint operator of $e(\gamma)$ with respect to any metric on $L$ , with or without the compactness or completeness assumptions on the base manifold.

Next we recall the harmonic theory in a local setting [Tak95]. Let $V$ be a bounded domain with smooth boundary $\partial V$ on $X$ . Assume that there is a smooth plurisubharmonic exhaustion function $r$ of $V$ , which is defined on a bigger neighbourhood $U$ with $\sup_{V}(|r|+|dr|)<\infty$ . In particular, $V=\{r<0\}$ and $dr\neq 0$ on $\partial V$ . The volume form $dS$ of the real hypersurface $\partial V$ is defined by $dS:=\ast(dr)/|dr|_{\omega}$ . Setting $\tau:=dS/|dr|_{\omega}$ we define the inner product on $\partial V$ by

[\alpha,\beta]_{\varphi,\omega}:=\int_{\partial V}\langle\alpha,\beta\rangle_{\varphi,\omega}\tau

for $\alpha,\beta\in A^{p,q}(\bar{V},L)_{\varphi,\omega}$ . Then by Stokes’ theorem we have the following:

\begin{split}(\bar{\partial}\alpha,\beta)_{\varphi,\omega}&=(\alpha,\bar{\partial}^{\ast}_{\varphi}\beta)_{\varphi,\omega}+[\alpha,e(\bar{\partial}r)^{\ast}\beta]_{\varphi,\omega},\\ (\partial_{\varphi}\alpha,\beta)_{\varphi,\omega}&=(\alpha,\partial^{\ast}_{\varphi}\beta)_{\varphi,\omega}+[\alpha,e(\partial r)^{\ast}\beta]_{\varphi,\omega}.\end{split}

(3.1)

The space of harmonic forms on $V$ is then defined as

\mathcal{H}^{n,q}(V,L,r):=\{\alpha\in A^{n,q}(V,L)_{\varphi,\omega};\bar{\partial}\alpha=\bar{\partial}^{\ast}_{\varphi}\alpha=e(\bar{\partial}r)^{\ast}\alpha=0\}.

Start from this space, K. Takegoshi generalised Kollar’s injectivity theorem to the Kähler setting. However, the metrics in our paper are not always smooth. Therefore we will further develop this theory next section so that it is applicable in our case.

In the end, we collect several formulas from [Tak95] for the later reference. The first one is the Calabi–Nakano–Vesentini formula:

\Box_{\varphi}=\overline{\Box}_{\varphi}+[i\Theta_{L,\varphi},\Lambda]\textrm{ for }\Box_{\varphi}:=\bar{\partial}\bar{\partial}^{\ast}_{\varphi}+\bar{\partial}^{\ast}_{\varphi}\bar{\partial}\quad\textrm{and}\quad\overline{\Box}_{\varphi}:=\partial_{\varphi}\partial^{\ast}_{\varphi}+\partial^{\ast}_{\varphi}\partial_{\varphi}.

(3.2)

Let $\psi$ be a real-valued smooth function on $X$ . Replacing the metric by $\varphi+\psi$ , we obtain the following variant:

\begin{split}\Box_{\varphi+\psi}&=\overline{\Box}_{\varphi+\psi}+[i\Theta_{L,\varphi}+i\partial\bar{\partial}\psi,\Lambda]\textrm{ for}\\ \Box_{\varphi+\psi}&:=\bar{\partial}\bar{\partial}^{\ast}_{\varphi+\psi}+\bar{\partial}^{\ast}_{\varphi+\psi}\bar{\partial}\quad\textrm{and}\quad\overline{\Box}_{\varphi+\psi}:=\partial_{\varphi+\psi}\partial^{\ast}_{\varphi+\psi}+\partial^{\ast}_{\varphi+\psi}\partial_{\varphi+\psi}.\end{split}

(3.3)

Donnelly and Xavier’s formula [DoX84] can be formulated as follows:

[\bar{\partial},e(\bar{\partial}\psi)^{\ast}]+[\partial^{\ast}_{\varphi},e(\partial\psi)]=[ie(\partial\bar{\partial}\psi),\Lambda].

(3.4)

3.2 The harmonic forms concerning singular metrics

Assume that $(L,\varphi)$ is a pseudo-effective line bundle such that $\varphi$ has analytic singularities [Dem12]. In particular, $\varphi$ is smooth on $V\setminus Z$ for a closed subvariety $Z$ . Note that $Z$ is at least of real codimension $2$ . So formula (3.1) is still valid on $V\setminus Z$ . The formulas (3.2)-(3.4) are established pointwise and thus make sense on arbitrary Kähler manifold such as $X$ , $V$ and $V\setminus Z$ .

We construct a complete Kähler metric $\omega_{l}$ on $V\setminus Z$ as follows: note that $V$ is exhausted by a smooth plurisubharmonic function $r$ , then by [Dem82] it is complete. Moreover, there exists a complete Kähler metric $\tilde{\omega}$ on $V\setminus Z$ . Let $\omega_{l}:=\omega+\frac{1}{l}\tilde{\omega}$ . Then $\omega_{l}$ is also complete on $V\setminus Z$ for all $l$ . Moreover, $\omega_{l_{2}}\geqslant\omega_{l_{1}}\geqslant\omega$ when $l_{2}\leqslant l_{1}$ . Let $\Box_{l}=\bar{\partial}\bar{\partial}^{\ast}_{\varphi}+\bar{\partial}^{\ast}_{\varphi}\bar{\partial}$ be the Laplacian operator associated to $\varphi,\omega_{l}$ . The harmonic form with respect to $\varphi$ is defined as

Definition 3.1.

Let $\alpha$ be a smooth $L$ -valued $(n,q)$ -form on $V$ with bounded $L^{2}$ -norm with respect to $\omega,\varphi$ . Assume that for every $l\gg 1$ , there exists $\alpha_{l}\in[\alpha|_{V\setminus Z}]$ such that

(1)

$\bar{\partial}\alpha_{l}=0$ , $\bar{\partial}^{\ast}_{\varphi}\alpha_{l}=0$ and $e(\bar{\partial}r)^{\ast}\alpha_{l}=0$ on $V\setminus Z$ ;
(2)

$\alpha_{l}\rightarrow\alpha|_{V\setminus Z}$ in the sense of $L^{2}$ -norm.

Then we call $\alpha$ a $\Box_{\varphi}$ -harmonic form. The space of all the $\Box_{\varphi}$ -harmonic forms is denoted by

\mathcal{H}^{n,q}(V,L\otimes\mathscr{I}(\varphi),r).

Here $\alpha_{l}\in[\alpha|_{V\setminus Z}]$ means that there exits an $L$ -valued $(n,q-1)$ -form $\beta_{l}$ on $V\setminus Z$ , such that $\alpha|_{V\setminus Z}=\alpha_{l}+\bar{\partial}\beta_{l}$ .

We then generalise several propositions in [Tak95] here.

Proposition 3.1.

We have the following conclusions:

1.

Assume $\alpha\in A^{n,q}(V\setminus Z,L)_{\varphi,\omega_{l}}$ satisfies $e(\bar{\partial}r)^{\ast}\alpha=0$ . Then $\alpha$ satisfies $\bar{\partial}\alpha=\bar{\partial}^{\ast}_{\varphi}\alpha=0$ if and only if $\bar{\partial}\ast\alpha=0$ and $(ie(\Theta_{L,\varphi}+\partial\bar{\partial}r)\Lambda\alpha,\alpha)_{\varphi,\omega_{l}}=0$ .
2.

$\mathcal{H}^{n,q}(V,L\otimes\mathscr{I}(\varphi),r)$ is independent of the choice of exhaustion function $r$ .

Proof.

The proof uses the same argument as Theorem 4.3 in [Tak95] with minor adjustment. So we only provide the necessary details. Let $V_{c}:=\{r<c\}$ for every $c\in(-\infty,0)$ .

(i) Take any regular value $c$ of $r$ . If $\bar{\partial}\alpha=\bar{\partial}^{\ast}_{\varphi}\alpha=0$ , by formulas (3.1) and (3.2) we obtain

\|\partial^{\ast}_{\varphi}\alpha\|^{2}_{\varphi,\omega_{l}}+(i\Theta_{L,\varphi}\Lambda\alpha,\alpha)_{\varphi,\omega_{l}}+[\partial^{\ast}_{\varphi}\alpha,e(\partial r)^{\ast}\alpha]_{\varphi,\omega_{l}}=0

on $V_{c}\setminus Z$ . Hence we obtain by (3.4)

\|\partial^{\ast}_{\varphi}\alpha\|^{2}_{\varphi,\omega_{l}}+(i\Theta_{L,\varphi}\Lambda\alpha,\alpha)_{\varphi,\omega_{l}}+[ie(\partial\bar{\partial}r)\Lambda\alpha,\alpha]_{\varphi,\omega_{l}}=0

on $V_{c}\setminus Z$ , which implies $\partial^{\ast}_{\varphi}\alpha=\ast\bar{\partial}\ast\alpha=0$ and $(i\Theta_{L,\varphi}\Lambda\alpha,\alpha)_{\varphi,\omega_{l}}=[ie(\partial\bar{\partial}r)\Lambda\alpha,\alpha]_{\varphi,\omega_{l}}=0$ by the plurisubharmonicity of $r$ . Apply (3.4) again we obtain

(ie(\partial\bar{\partial}r)\Lambda\alpha,\alpha)_{\varphi,\omega_{l}}=(ie(\partial r)\partial^{\ast}_{\varphi}\alpha,\alpha)_{\varphi,\omega_{l}}=0.

The necessity is then proved as $c$ varies.

If $\bar{\partial}\ast\alpha=0$ and $(ie(\Theta_{L,\varphi}+\partial\bar{\partial}r)\Lambda\alpha,\alpha)_{\varphi,\omega_{l}}=0$ , by the plurisubharmonicity of $r$

(i\Theta_{L,\varphi}\Lambda\alpha,\alpha)_{\varphi,\omega_{l}}=0\quad\textrm{and}\quad(ie(\partial\bar{\partial}r)\Lambda\alpha,\alpha)_{\varphi,\omega_{l}}=0

on $V\setminus Z$ . By (3.2) and (3.4) we obtain:

(\Box_{l}\alpha,\alpha)_{\varphi,\omega_{l}}=0\quad\textrm{and}\quad(e(\bar{\partial}r)^{\ast}\bar{\partial}\alpha,\alpha)_{\varphi,\omega_{l}}=0

on $V_{c}\setminus Z$ . Hence by (3.1) we obtain:

0=(\Box_{l}\alpha,\alpha)_{\varphi,\omega_{l}}=\|\bar{\partial}\alpha\|^{2}_{\varphi,\omega_{l}}+\|\bar{\partial}^{\ast}_{\varphi}\alpha\|^{2}_{\varphi,\omega_{l}}

on $V_{c}\setminus Z$ , which implies the sufficiency as $c$ varies.

(ii) Let $\tau$ be an arbitrary smooth plurisubharmonic function defined on a bigger neighbourhood $U$ containing $V$ . Given

\alpha\in\mathcal{H}^{n,q}(V,L\otimes\mathscr{I}(\varphi),r),

there exits $\{\alpha_{l}\}$ on $V\setminus Z$ with $\bar{\partial}\alpha_{l}=\bar{\partial}^{\ast}_{\varphi}\alpha_{l}=e(\bar{\partial}r)^{\ast}\alpha_{l}=0$ , and is convergent to $\alpha|_{V\setminus Z}$ by definition. Formula (3.4) implies that $\bar{\partial}(e(\bar{\partial}\tau)^{\ast}\alpha_{l})+e(\partial\tau)\partial^{\ast}_{\varphi}\alpha_{l}=ie(\partial\bar{\partial}\tau)\Lambda\alpha_{l}$ on $V\setminus Z$ . Therefore

\begin{split}&(ie(\partial\bar{\partial}\tau)\Lambda\alpha_{l},\alpha_{l})_{\varphi-\tau,\omega_{l}}\\ =&(\bar{\partial}(e(\bar{\partial}\tau)^{\ast}\alpha_{l}),\alpha_{l})_{\varphi-\tau,\omega_{l}}+(e(\partial\tau)\partial^{\ast}_{\varphi}\alpha_{l},\alpha_{l})_{\varphi-\tau,\omega_{l}}\\ =&(e(\bar{\partial}\tau)^{\ast}\alpha_{l},\bar{\partial}^{\ast}_{\varphi-\tau}\alpha_{l})_{\varphi-\tau,\omega_{l}}+(e(\partial\tau)\partial^{\ast}_{\varphi}\alpha_{l},\alpha_{l})_{\varphi-\tau,\omega_{l}}\\ =&-\|e(\bar{\partial}\tau)^{\ast}\alpha_{l}\|^{2}_{\varphi-\tau,\omega_{l}}+(e(\partial\tau)\partial^{\ast}_{\varphi}\alpha_{l},\alpha_{l})_{\varphi-\tau,\omega_{l}}\end{split}

on $V_{c}\setminus Z$ for any non-critical value $c$ of $r$ . Note $\partial^{\ast}_{\varphi}\alpha_{l}=0$ by (i). We then obtain that

(ie(\partial\bar{\partial}\tau)\Lambda\alpha_{l},\alpha_{l})_{\varphi-\tau,\omega_{l}}=-\|e(\bar{\partial}\tau)^{\ast}\alpha_{l}\|^{2}_{\varphi-\tau,\omega_{l}}\textrm{ on }V_{c}\setminus Z.

Notice that $\tau$ is plurisubharmonic, as $c$ varies we actually have

(ie(\partial\bar{\partial}\tau)\Lambda\alpha_{l},\alpha_{l})_{\varphi-\tau,\omega_{l}}=\|e(\bar{\partial}\tau)^{\ast}\alpha_{l}\|^{2}_{\varphi-\tau,\omega_{l}}=0\textrm{ on }V\setminus Z.

Obviously it is equivalent to say that

(ie(\partial\bar{\partial}\tau)\Lambda\alpha_{l},\alpha_{l})_{\varphi,\omega_{l}}=\|e(\bar{\partial}\tau)^{\ast}\alpha_{l}\|^{2}_{\varphi,\omega_{l}}=0\textrm{ on }V\setminus Z.

Combine with (i), we eventually obtain that

\mathcal{H}^{n,q}(V,L\otimes\mathscr{I}(\varphi),r)=\mathcal{H}^{n,q}(V,L\otimes\mathscr{I}(\varphi),r+\tau)

for any smooth plurisubharmonic $\tau$ , hence the desired conclusion. ∎

3.3 The Hodge-type isomorphism

In this section, we return to the relative setting. Let $f:X\rightarrow Y$ be a proper, locally Kähler morphism from a complex manifold $X$ to a reduced, irreducible analytic space $Y$ . Every connected component of $X$ is mapped surjectively to $Y$ . Let $l$ be the dimension of a general fibre $F$ of $f$ .

Suppose that $(L,\varphi)$ is a pseudo-effective line bundle on $X$ with analytic singularities. Let $Z$ be the closed subvariety such that $\varphi$ is smooth on $X\setminus Z$ . Let $\{U\}$ be a Stein covering of $Y$ , then $f^{-1}(U)$ is Kähler. In particular, we could construct complete Kähler metrics $\omega_{l}$ on $f^{-1}(U)\setminus Z$ as is shown before.

Let $r_{U}$ be a smooth strictly plurisubharmonic exhaustion function on $U$ . In particular,

\sup_{U}(|r_{U}|+|dr_{U}|)<\infty

after shrinking $U$ if necessary. Let

\mathcal{H}^{n,q}(f^{-1}(U),L\otimes\mathscr{I}(\varphi),f^{\ast}r_{U})

be the harmonic space in Definition 3.1. We have

Proposition 3.2.

1.

$\mathcal{H}^{n,q}(f^{-1}(U),L\otimes\mathscr{I}(\varphi),f^{\ast}r_{U})\simeq H^{q}(f^{-1}(U),K_{X}\otimes L\otimes\mathscr{I}(\varphi))$ .

If $V$ is a Stein open subset of $U$ provided with a smooth strictly plurisubharmonic exhaustion function $r_{V}$ , then the restriction map

\mathcal{H}^{n,q}(f^{-1}(U),L\otimes\mathscr{I}(\varphi),f^{\ast}r_{U})\rightarrow\mathcal{H}^{n,q}(f^{-1}(V),\mathscr{I}(\varphi),f^{\ast}r_{V})

is well-defined, and further the following diagram is commutative:

The morphism $S^{q}_{U}$ will be verified during the proof.

Proof.

(i) When $\varphi$ is smooth, it is nothing but [Tak95], Theorem 5.2. Whereas our $\varphi$ is merely smooth outside $Z$ , then the idea is to apply the similar argument on $f^{-1}(U)\setminus Z$ to obtain the desired conclusion.

More precisely, let $L^{n,q}_{(2)}(f^{-1}(U),L)_{\varphi,\omega}$ be the space of $L$ -valued $(n,q)$ -forms on $f^{-1}(U)$ which are $L^{2}$ -bounded with respect to $\varphi,\omega$ . Let $[\alpha]\in H^{q}(f^{-1}(U),K_{X}\otimes L\otimes\mathscr{I}(\varphi))$ . We use the de Rham–Weil isomorphism

H^{q}(f^{-1}(U),K_{X}\otimes L\otimes\mathscr{I}(\varphi))\simeq\frac{\mathrm{Ker}(\bar{\partial}:L^{n,q}_{(2)}(f^{-1}(U),L)_{\varphi,\omega}\rightarrow L^{n,q+1}_{(2)}(f^{-1}(U),L)_{\varphi,\omega})}{\mathrm{Im}(\bar{\partial}:L^{n,q-1}_{(2)}(f^{-1}(U),L)_{\varphi,\omega}\rightarrow L^{n,q}_{(2)}(f^{-1}(U),L)_{\varphi,\omega})}

to represent it by a $\bar{\partial}$ -closed $L$ -valued $(n,q)$ -form $\alpha$ with $\|\alpha\|_{\varphi,\omega}<\infty$ .

Let $L^{n,q}_{(2)}(f^{-1}(U)\setminus Z,L)_{\varphi,\omega_{l}}$ be the space of $L$ -valued $(n,q)$ -forms on $f^{-1}(U)\setminus Z$ which are $L^{2}$ -bounded with respect to $\varphi,\omega_{l}$ . Since $\varphi$ is smooth on $f^{-1}(U)\setminus Z$ , we have the orthogonal decomposition

L^{n,q}_{(2)}(f^{-1}(U)\setminus Z,L)_{\varphi,\omega_{l}}=\overline{\mathrm{Im}\bar{\partial}}\bigoplus\mathcal{H}^{n,q}_{\varphi,\omega_{l}}(L)\bigoplus\overline{\mathrm{Im}\bar{\partial}^{\ast}_{\varphi}},

where

\mathcal{H}^{n,q}_{\varphi,\omega_{l}}(L)=\{\alpha\in\textrm{Dom}\bar{\partial}\cap\textrm{Dom}\bar{\partial}^{\ast}_{\varphi};\bar{\partial}\alpha=0,\bar{\partial}^{\ast}_{\varphi}\alpha=0\}

with

\textrm{Dom}\bar{\partial}=\{\alpha\in L^{n,q}_{(2)}(f^{-1}(U)\setminus Z,L)_{\varphi,\omega_{l}};\bar{\partial}\alpha\in L^{n,q+1}_{(2)}(f^{-1}(U)\setminus Z,L)_{\varphi,\omega_{l}}\}

and

\textrm{Dom}\bar{\partial}^{\ast}_{\varphi}=\{\alpha\in L^{n,q}_{(2)}(f^{-1}(U)\setminus Z,L)_{\varphi,\omega_{l}};\bar{\partial}^{\ast}_{\varphi}\alpha\in L^{n,q-1}_{(2)}(f^{-1}(U)\setminus Z,L)_{\varphi,\omega_{l}}\}.

Moreover, for any $u\in\overline{\mathrm{Im}\bar{\partial}}\cap A^{n,q}(f^{-1}(U)\setminus Z,L)$ , there is $v\in A^{n,q-1}(f^{-1}(U)\setminus Z,L)$ such that $u=\bar{\partial}v$ . It is essentially due to the fact that $f$ is proper and

H^{q}(f^{-1}(U)\setminus Z,K_{X}\otimes L)

is a separated topological space. We recommend [Tak95], Proposition 4.6 or [Fuj12], Claim 1 for references. Now we set $\alpha_{l}:=$ the orthogonal projection of $\alpha|_{f^{-1}(U)\setminus Z}$ onto $\mathcal{H}^{n,q}_{\varphi,\omega_{l}}(L)$ , which is smooth by the regularization theorem for elliptic operators of second order. Hence

\alpha|_{f^{-1}(U)\setminus Z}-\alpha_{l}\in\overline{\mathrm{Im}\bar{\partial}}\cap A^{n,q}(f^{-1}(U)\setminus Z,L).

Then there is $\beta_{l}\in A^{n,q-1}(f^{-1}(U)\setminus Z,L)$ such that $\alpha|_{f^{-1}(U)\setminus Z}=\alpha_{l}+\bar{\partial}\beta_{l}$ . Moreover, the representative

\alpha_{l}\in[\alpha|_{f^{-1}(U)\setminus Z}]

minimizes the $L^{2}$ -norm defined by $\varphi$ and $\omega_{l}$ . Now we claim that $e(\bar{\partial}f^{\ast}r_{U})^{\ast}\alpha_{l}=0$ .

In order to prove this claim, we first recall the following formula in [Tak95], Theorem 2.2. Let $\psi$ be a smooth plurisubharmonic function on $f^{-1}(U)\setminus Z$ such that

\sup_{f^{-1}(U)\setminus Z}(|\psi|+|d\psi|)<\infty

and let $\eta=e^{\psi}$ , then

\begin{split}&\|\sqrt{\eta}(\bar{\partial}+e(\bar{\partial}\psi))\beta\|^{2}_{\varphi,\omega_{l}}+\|\sqrt{\eta}\bar{\partial}^{\ast}_{\varphi}\beta\|^{2}_{\varphi,\omega_{l}}\\ =&\|\sqrt{\eta}(\partial^{\ast}_{\varphi}-e(\partial\psi))^{\ast}\beta\|^{2}_{\varphi,\omega_{l}}+(i\eta(\Theta_{L,\varphi}+\partial\bar{\partial}\psi)\Lambda\beta,\beta)_{\varphi,\omega_{l}}\end{split}

(3.5)

for any $\beta\in\textrm{Dom}\bar{\partial}\cap\textrm{Dom}\bar{\partial}^{\ast}_{\varphi}\subseteq L^{n,q}_{(2)}(f^{-1}(U)\setminus Z,L)_{\varphi,\omega_{l}}$ . We apply (3.5) with $\beta=\alpha_{l}$ and $\psi=0$ , then

0=\|\partial^{\ast}_{\varphi}\alpha_{l}\|^{2}_{\varphi,\omega_{l}}+(i\Theta_{L,\varphi}\Lambda\alpha_{l},\alpha_{l})_{\varphi,\omega_{l}}.

Thus, we obtain that

\partial^{\ast}_{\varphi}\alpha_{l}=0\quad\textrm{and}\quad(i\Theta_{L,\varphi}\Lambda\alpha_{l},\alpha_{l})_{\varphi,\omega_{l}}=0.

Apply (3.5) again with $\beta=\alpha_{l}$ and $\psi=f^{\ast}r_{U}$ , we can obtain

\|e(\bar{\partial}f^{\ast}r_{U})\alpha_{l}\|^{2}_{\varphi-f^{\ast}r_{U},\omega_{l}}=\|e(\partial f^{\ast}r_{U})^{\ast}\alpha_{l}\|^{2}_{\varphi-f^{\ast}r_{U},\omega_{l}}+(ie(\partial\bar{\partial}f^{\ast}r_{U})\Lambda\alpha_{l},\alpha_{l})_{\varphi-f^{\ast}r_{U},\omega_{l}}.

(3.6)

On the other hand, it is easy to deduce the following equation:

\|e(\partial f^{\ast}r_{U})^{\ast}\alpha_{l}\|^{2}_{\varphi-f^{\ast}r_{U},\omega_{l}}=\|e(\bar{\partial}f^{\ast}r_{U})\alpha_{l}\|^{2}_{\varphi-f^{\ast}r_{U},\omega_{l}}+\|e(\bar{\partial}f^{\ast}r_{U})^{\ast}\alpha_{l}\|^{2}_{\varphi-f^{\ast}r_{U},\omega_{l}}

on $f^{-1}(U)\setminus Z$ from the Kähler identity:

e(\theta)^{\ast}=i[e(\bar{\theta}),\Lambda]

for $\theta\in A^{1,0}(f^{-1}(U)\setminus Z)$ . Finally we obtain that $e(\bar{\partial}f^{\ast}r_{U})^{\ast}\alpha_{l}=0$ from the plurisubharmonicity of $f^{\ast}r_{U}$ . The claim is then proved. It remains to show that $\{\alpha_{l}\}$ is convergent to $\hat{\alpha}|_{f^{-1}(U)\setminus Z}$ for some smooth $L$ -valued $(n,q)$ -form $\hat{\alpha}$ on $f^{-1}(U)$ .

Fix $l_{0}$ . For any $l\geqslant l_{0}$ we have

\|\ast\alpha_{l}\|_{\varphi,\omega_{l_{0}}}\leqslant\|\ast\alpha_{l}\|_{\varphi,\omega_{l}}=\|\alpha_{l}\|_{\varphi,\omega_{l}}\leqslant\|\alpha|_{f^{-1}(U)\setminus Z}\|_{\varphi,\omega_{l}}\leqslant\|\alpha\|_{\varphi,\omega}<\infty.

Here we use the fact that $\alpha_{l}$ minimizes the $L^{2}$ -norm $\|\cdot\|_{\varphi,\omega_{l}}$ . It means that $\ast\alpha_{l}$ has uniformly bounded $L^{2}$ -norm with respect to $\|\cdot\|_{\varphi,\omega_{l_{0}}}$ on $f^{-1}(U)\setminus Z$ . On the other hand, since

0=\partial^{\ast}_{\varphi}\alpha_{l}=\ast\bar{\partial}\ast\alpha_{l},

$\ast\alpha_{l}$ is a holomorphic $L$ -valued $(n-q,0)$ -form on $f^{-1}(U)\setminus Z$ . Therefore, it extends as $\theta_{l}$ to the whole space by classic $L^{2}$ -extension theorem [Ohs02]. Due to the fact that $\|\theta_{l}\|_{\varphi,\omega_{l_{0}}}$ is uniformly bounded, $\{\theta_{l}\}$ converges to some $\theta$ in the sense of $L^{2}$ -norm $\|\cdot\|_{\varphi,\omega_{l_{0}}}$ . Since $\theta_{l}$ is holomorphic, so is $\theta$ . It means that $\theta_{l}$ is actually convergent to $\theta$ with respect to any $L^{2}$ -norm. Now, since $\|\theta\|_{\varphi,\omega_{l_{0}}}\leqslant\|\alpha\|_{\varphi,\omega}$ , we actually have $\|\theta\|_{\varphi,\omega}\leqslant\|\alpha\|_{\varphi,\omega}$ as $l_{0}$ tends to infinity. In summary, we have defined a morphism as follows so far:

\begin{split}S^{q}_{U}:H^{q}(f^{-1}(U),K_{X}\otimes L\otimes\mathscr{I}(\varphi))&\rightarrow H^{0}(f^{-1}(U),\Omega^{n-q}_{X}\otimes L\otimes\mathscr{I}(\varphi))\\ [\alpha]&\mapsto\theta.\end{split}

Now let $c_{n-q}=(i)^{(n-q)^{2}}$ , and let $\gamma_{l}=c_{n-q}\frac{\omega^{q}_{l}}{q!}\wedge\theta_{l}$ . Obviously we have

c_{n-q}\frac{\omega^{q}}{q!}\wedge\theta=\lim_{l\rightarrow\infty}\gamma_{l}=\lim_{l\rightarrow\infty}c_{n-q}\frac{\omega^{q}_{l}}{q!}\wedge\ast\alpha_{l}=\lim_{l\rightarrow\infty}\alpha_{l}\textrm{ on }f^{-1}(U)\setminus Z.

(3.7)

Let $\hat{\alpha}:=c_{n-q}\frac{\omega^{q}}{q!}\wedge\theta$ , then it is a smooth $L$ -valued $(n,q)$ -form on $f^{-1}(U)$ with bounded $L^{2}$ -norm $\|\cdot\|_{\varphi,\omega}$ . Now

\hat{\alpha}\in\mathcal{H}^{n,q}(f^{-1}(U),L\otimes\mathscr{I}(\varphi),f^{\ast}r_{U})

by definition. We denote this morphism by $i([\alpha])=\hat{\alpha}$ .

Conversely, for an $\alpha\in\mathcal{H}^{n,q}(f^{-1}(U),L\otimes\mathscr{I}(\varphi),f^{\ast}r_{U})$ , by definition there exists an

\alpha_{l}\in[\alpha|_{f^{-1}(U)\setminus Z}]

with $\alpha_{l}\in\mathcal{H}^{n,q}_{\varphi,\omega_{l}}(L)$ for every $l$ and $\lim\alpha_{l}=\alpha|_{f^{-1}(U)\setminus Z}$ . In particular, $\bar{\partial}\alpha_{l}=0$ . So

\bar{\partial}(\alpha|_{f^{-1}(U)\setminus Z})=0.

Since $\alpha$ is smooth, we actually have $\bar{\partial}\alpha=0$ on $f^{-1}(U)$ . It then defines a cohomology class

[\alpha]\in H^{q}(f^{-1}(U),K_{X}\otimes L\otimes\mathscr{I}(\varphi)).

We denote this morphism by $j(\alpha)=[\alpha]$ . It is easy to verify that $i\circ j=\textrm{id}$ and $j\circ i=\textrm{id}$ . The proof is finished.

(ii) Let $\alpha\in\mathcal{H}^{n,q}(f^{-1}(U),L\otimes\mathscr{I}(\varphi),f^{\ast}r_{U})$ , and let $\{\alpha_{l}\}$ be the sequence on $f^{-1}(U)\setminus Z$ that is convergent to $\alpha|_{f^{-1}(U)\setminus Z}$ . Obviously

\alpha_{l}|_{f^{-1}(V)\setminus Z}\in\mathcal{H}^{n,q}_{\varphi,\omega_{l}}(L)\subseteq L^{n,q}_{(2)}(f^{-1}(V)\setminus Z,L)_{\varphi,\omega_{l}}.

In order to show the restriction map is well-defined, we only need to prove that

e(\bar{\partial}f^{\ast}r_{V})^{\ast}\alpha_{l}|_{f^{-1}(V)\setminus Z}=0.

However, it is nothing but repeat to argument in (i). Technically, apply (3.5) on $f^{-1}(V)\setminus Z$ we obtain that

\partial^{\ast}_{\varphi}\alpha_{l}=0\quad\textrm{and}\quad(i\Theta_{L,\varphi}\Lambda\alpha_{l},\alpha_{l})_{\varphi,\omega_{l}}=0.

Apply (3.5) again with $\beta=\alpha_{l}$ and $\psi=f^{\ast}r_{V}$ , we can obtain

\begin{split}\|e(\bar{\partial}f^{\ast}r_{V})\alpha_{l}\|^{2}_{\varphi-f^{\ast}r_{V},\omega_{l}}=\|e(\partial f^{\ast}r_{V})^{\ast}\alpha_{l}\|^{2}_{\varphi-f^{\ast}r_{V},\omega_{l}}+(i\eta e(\partial\bar{\partial}f^{\ast}r_{V})\Lambda\alpha_{l},\alpha_{l})_{\varphi,\omega_{l}}.\end{split}

(3.8)

Combine with the following equation:

\|e(\partial f^{\ast}r_{V})^{\ast}\alpha_{l}\|^{2}_{\varphi-f^{\ast}r_{V},\omega_{l}}=\|e(\bar{\partial}f^{\ast}r_{V})\alpha_{l}\|^{2}_{\varphi-f^{\ast}r_{V},\omega_{l}}+\|e(\bar{\partial}f^{\ast}r_{V})^{\ast}\alpha_{l}\|^{2}_{\varphi-f^{\ast}r_{V},\omega_{l}},

finally we obtain that $e(\bar{\partial}f^{\ast}r_{V})^{\ast}\alpha_{l}=0$ from the plurisubharmonicity of $f^{\ast}r_{V}$ . Then everything is intuitive due to the discussions before. ∎

The data

\{\mathcal{H}^{n,q}(f^{-1}(U),L\otimes\mathscr{I}(\varphi),f^{\ast}r_{U}),i^{U}_{V}\}

with the restriction morphisms

i^{U}_{V}:\mathcal{H}^{n,q}(f^{-1}(U),L\otimes\mathscr{I}(\varphi),f^{\ast}r_{U})\rightarrow\mathcal{H}^{n,q}(f^{-1}(V),L\otimes\mathscr{I}(\varphi),f^{\ast}r_{V}),

$(V,r_{V})\subset(U,r_{U})$ , yields a presheaf [Har77] on $Y$ by Proposition 3.2, (ii). We denote the associated sheaf by $f_{\ast}\mathcal{H}^{n,q}(L\otimes\mathscr{I}(\varphi))$ . Since

R^{q}f_{\ast}(K_{X}\otimes L\otimes\mathscr{I}(\varphi))

is defined as the sheaf associated with the presheaf

U\rightarrow H^{q}(f^{-1}(U),K_{X}\otimes L\otimes\mathscr{I}(\varphi)),

the sheaf $f_{\ast}\mathcal{H}^{n,q}(L\otimes\mathscr{I}(\varphi))$ is isomorphic to $R^{q}f_{\ast}(K_{X}\otimes L\otimes\mathscr{I}(\varphi))$ by Proposition 3.2, (i).

Remark 3.1.

The whole harmonic theory could even be established for a pseudo-effective line bundle $(L,\varphi)$ with following property: there exists integers $k_{0}$ , $m$ and sections $s_{1},...,s_{m}\in L^{k_{0}}$ such that

\sup_{X}(\sum^{m}_{i=1}|s_{i}|^{2}e^{-\varphi})<\infty.

Readers may refer to [Wu20] for more details.

Remark 3.2.

Everything in this section works smoothly after tensoring with $E(h)$ provided that $(E,h)$ is a tame Nakano semi-positive vector bundle on $X^{o}$ . Here $X^{o}$ is a Zariski open subset of $X$ . We list the major adjustments along with brief explanations here.

(1)

In Definition 3.1, $V\setminus Z$ is replaced by $(V\setminus Z)\cap X^{o}$ , which is still complete for the same reason. The $L^{2}$ -norm is defined by $\omega,\varphi,h$ ;
(2)

Since $X^{o}$ is Zariski open, formulas (3.1)-(3.5) still hold except that the curvature term is exchanged as $i\Theta_{L\otimes E,\varphi,h}$ . It is semi-positive in the sense of Nakano hence the argument in Propositions 3.1 and 3.2 extends with no difficulties.
(3)

$f_{\ast}\mathcal{H}^{n,q}(L\otimes\mathscr{I}(\varphi))$ is replaced by $f_{\ast}\mathcal{H}^{n,q}(L\otimes E(h\cdot e^{-\varphi}))$ , and $R^{q}f_{\ast}(K_{X}\otimes L\otimes\mathscr{I}(\varphi))$ is replaced by $R^{q}f_{\ast}(K_{X}\otimes L\otimes E(h\cdot e^{-\varphi}))$ .

Now we will directly use the following isomorphism:

R^{q}f_{\ast}(K_{X}\otimes L\otimes E(h\cdot e^{-\varphi}))\simeq f_{\ast}\mathcal{H}^{n,q}(L\otimes E(h\cdot e^{-\varphi})).

(3.9)

This should not lead to any confusion.

In the end, remember that for a relative asymptotic multiplier ideal sheaf $\mathscr{I}(f,\|L\|)$ , we can always associate a collection of singular metrics $\{f^{-1}(U),\varphi_{U}\}$ with algebraic singularities [Dem12] on $L$ , such that

i\Theta_{L,\varphi_{U}}\geqslant 0\quad\textrm{and}\quad\mathscr{I}(\varphi_{U})=\mathscr{I}(f,\|L\|)\textrm{ on }f^{-1}(U).

After replacing the $\varphi$ before by $\{f^{-1}(U),\varphi_{U}\}$ , the isomorphism (3.9) is preserved.

Then we furthermore assume that there exists a section $s$ of some multiple $L^{m}$ such that $\{s=0\}\subseteq X^{o}$ . At this time $(V\setminus Z)\cap X^{o}=V\cap X^{o}$ , and $\varphi_{U}$ is smooth outside of $X^{o}$ . As a consequence, we have

E(h\cdot e^{-\varphi_{U}})=E(h)\otimes\mathscr{I}(\varphi_{U}),

and (3.9) is reformulated as

Proposition 3.3 ((Hodge-type isomorphism)).

Assume that there exists a section $s$ of some multiple $L^{m}$ such that $\{s=0\}\subseteq X^{o}$ . Then

R^{q}f_{\ast}(K_{X}\otimes L\otimes E(h)\otimes\mathscr{I}(f,\|L\|))\simeq f_{\ast}\mathcal{H}^{n,q}(L\otimes E(h)\otimes\mathscr{I}(f,\|L\|)).

4 Injectivity theorem and vanishing theorem

We first prove Theorem 1.3, i.e. a Kollár-type injectivity and torsion-freeness theorem. Note that throughout the rest part of this paper, $f:X\rightarrow Y$ is a proper, locally Kähler morphism from a complex manifold $X$ to a reduced, irreducible analytic space $Y$ . Every connected component of $X$ is mapped surjectively to $Y$ . Let $l$ be the dimension of a general fibre $F$ of $f$ . $X^{o}$ is a Zariski open subset of $X$ , and $(E,h)$ is a tame, Nakano semi-positive vector bundle over $X^{o}$ . Denote the fact that $i\Theta_{E,h}$ is semi-positive in the sense of Nakano by $i\Theta_{E,h}\geqslant_{\textrm{Nak}}0$ .

4.1 A Kollár-type injectivity theorem

Theorem 4.1 ((=Theorem 1.3)).

\Phi:R^{q}f_{\ast}(K_{X}\otimes L\otimes E(h)\otimes\mathscr{I}(f,\|L\|))\rightarrow R^{q}f_{\ast}(K_{X}\otimes L^{m}\otimes E(h)\otimes\mathscr{I}(f,\|L^{m}\|))

is well-defined and injective for any $q\geqslant 0$ . In particular, $R^{q}f_{\ast}(K_{X}\otimes L\otimes E(h)\otimes\mathscr{I}(f,\|L\|))$ is torsion-free for every $q$ .

Proof.

Let $\{U\}$ be a Stein covering of $Y$ . Let $r_{U}$ be a smooth strictly plurisubharmonic exhaustion function on $U$ . From the discussion in Sect.2.1, there exits a collection of singular metrics

\varphi=\{f^{-1}(U),\varphi_{U}\}

with algebraic singularities, such that

\mathscr{I}(f,\|L\|)=\mathscr{I}(\varphi_{U}),\mathscr{I}(f,\|L^{m-1}\|)=\mathscr{I}((m-1)\varphi_{U})\textrm{ and }\mathscr{I}(f,\|L^{m}\|)=\mathscr{I}(m\varphi_{U})\textrm{ on }f^{-1}(U).

Then in view of Proposition 3.3 it is left to prove that

\otimes s:f_{\ast}\mathcal{H}^{n,q}(L\otimes E(h)\otimes\mathscr{I}(\varphi))\rightarrow f_{\ast}\mathcal{H}^{n,q}(L^{m}\otimes E(h)\otimes\mathscr{I}(m\varphi))

is well-defined and injective.

Let $\alpha\in\mathcal{H}^{n,q}(f^{-1}(U),L\otimes E(h)\otimes\mathscr{I}(\varphi_{U}),f^{\ast}r_{U})$ , and let $Z$ be the closed subvariety such that $\varphi$ is smooth on $f^{-1}(U)\setminus Z$ . Then $Z\cap X^{o}=\emptyset$ by assumption. By definition, there exists a sequence $\{\alpha_{l}\}$ such that

\alpha_{l}\in\mathcal{H}^{n,q}_{\varphi_{U},h,\omega_{l}}(f^{-1}(U)\cap X^{o},L\otimes E),\alpha_{l}\in[\alpha|_{f^{-1}(U)\cap X^{o}}],e(\bar{\partial}f^{\ast}r_{U})^{\ast}\alpha_{l}=0

and $\lim\alpha_{l}=\alpha|_{f^{-1}(U)\cap X^{o}}$ in the sense of $L^{2}$ -topology. Apply formula (3.5) to

\alpha_{l}\textrm{ on }f^{-1}(U)\cap X^{o},

we obtain

\begin{split}0=&\|\bar{\partial}\alpha_{l}\|^{2}_{\varphi_{U},h,\omega_{l}}+\|\bar{\partial}^{\ast}_{\varphi_{U},h}\alpha_{l}\|^{2}_{\varphi_{U},h,\omega_{l}}\\ =&\|\partial^{\ast}_{\varphi_{U},h}\alpha_{l}\|^{2}_{\varphi_{U},h,\omega_{l}}+(i\Theta_{L\otimes E,\varphi_{U},h}\Lambda\alpha_{l},\alpha_{l})_{\varphi_{U},h,\omega_{l}}.\end{split}

Remember that $i\Theta_{L\otimes E,\varphi_{U},h}\geqslant_{\textrm{Nak}}0$ . Thus, $\partial_{\varphi_{U}}^{\ast}\alpha_{l}=0$ and

(i\Theta_{L\otimes E,\varphi_{U},h}\Lambda\alpha_{l},\alpha_{l})_{\varphi_{U},h,\omega_{l}}=0.

Now for an

s\in H^{0}(X,L^{m-1}),

certainly we have $\bar{\partial}(s\alpha)=0$ and $e(\bar{\partial}f^{\ast}r_{U})^{\ast}(s\alpha_{l})=0$ . Let $\mathfrak{a}(f,|L|)$ be the base-ideal of $|L|$ relative to $f$ , so $s\in\mathfrak{a}(f,|L^{m-1}|)$ . Then

[s\alpha]\in H^{q}(f^{-1}(U),K_{X}\otimes L^{m}\otimes E(h)\otimes\mathscr{I}(m\varphi_{U}))

by Proposition 2.1. Then due to Proposition 3.2, there exists a sequence $\{\beta_{l}\}$ on $f^{-1}(U)\cap X^{o}$ such that $\bar{\partial}\beta_{l}=\bar{\partial}^{\ast}_{m\varphi_{U},h}\beta_{l}=e(\bar{\partial}f^{\ast}r_{U})^{\ast}\beta_{l}=0$ and $\beta_{l}\in[(s\alpha)|_{f^{-1}(U)\cap X^{o}}]$ . It is left to prove that $\beta_{l}=s\alpha_{l}$ . Indeed, since $\beta_{l},s\alpha_{l}\in[(s\alpha)|_{f^{-1}(U)\cap X^{o}}]$ , there exits an $L^{m}\otimes E$ -valued $(n,q-1)$ -form $\gamma_{l}$ on $f^{-1}(U)\cap X^{o}$ such that $s\alpha_{l}=\beta_{l}+\bar{\partial}\gamma_{l}$ . Now apply (3.1) and (3.2) to $s\alpha_{l}$ on $f^{-1}(U)\cap X^{o}$ after shrinking $U$ if necessary, we obtain that

\begin{split}\|\bar{\partial}^{\ast}_{m\varphi_{U},h}(s\alpha_{l})\|^{2}_{m\varphi_{U},h,\omega_{l}}=&\|\partial^{\ast}_{m\varphi_{U},h}(s\alpha_{l})\|^{2}_{m\varphi_{U},h,\omega_{l}}+(i\Theta_{L^{m}\otimes E,m\varphi_{U},h}\Lambda(s\alpha_{l}),s\alpha_{l})_{m\varphi_{U},h,\omega_{l}}\\ &+[\partial^{\ast}_{m\varphi_{U},h}(s\alpha_{l}),e(\partial f^{\ast}r_{U})^{\ast}(s\alpha_{l})]_{m\varphi_{U},h,\omega_{l}}.\end{split}

Note $\partial^{\ast}_{m\varphi_{U},h}(s\alpha_{l})=-\ast\bar{\partial}\ast(s\alpha_{l})=-s\ast\bar{\partial}\ast\alpha_{l}=s\partial^{\ast}_{\varphi_{U},h}\alpha_{l}=0$ . On the other hand,

\begin{split}0\leqslant&(i\Theta_{L^{m}\otimes E,m\varphi_{U},h}\Lambda(s\alpha_{l}),s\alpha_{l})_{m\varphi_{U},h,\omega_{l}}\\ \leqslant&\sup_{X}(|s|^{2}e^{-(m-1)\varphi_{U}})(i\Theta_{L^{m}\otimes E,\varphi_{U},h}\Lambda\alpha_{l},\alpha_{l})_{\varphi_{U},h,\omega_{l}}\\ \leqslant&m\sup_{X}(|s|^{2}e^{-(m-1)\varphi_{U}})(i\Theta_{L\otimes E,\varphi_{U},h}\Lambda\alpha_{l},\alpha_{l})_{\varphi_{U},h,\omega_{l}}\\ =&0.\end{split}

Since $\sup_{X}(|s|^{2}e^{-(m-1)\varphi_{U}})$ is obviously finite, we obtain that

(i\Theta_{L^{m}\otimes E,m\varphi_{U},h}\Lambda(s\alpha_{l}),s\alpha_{l})_{m\varphi_{U},h,\omega_{l}}=0.

In summary,

\|\bar{\partial}^{\ast}_{m\varphi_{U},h}(s\alpha_{l})\|^{2}_{m\varphi_{U},h,\omega_{l}}=0.

Then we have

\begin{split}&\|\bar{\partial}^{\ast}_{m\varphi_{U},h}\bar{\partial}\gamma_{l}\|^{2}_{m\varphi_{U},h,\omega_{l}}\\ =&\|\bar{\partial}^{\ast}_{m\varphi_{U},h}(s\alpha_{l}-\beta_{l})\|^{2}_{m\varphi_{U},h,\omega_{l}}\\ =&0.\end{split}

In other words, $\bar{\partial}^{\ast}_{m\varphi_{U},h}\bar{\partial}\gamma_{l}=0$ . Observe that $e(\bar{\partial}f^{\ast}r_{U})^{\ast}\bar{\partial}\gamma_{l}=e(\bar{\partial}f^{\ast}r_{U})^{\ast}(s\alpha_{l}-\beta_{l})=0$ . Therefore by (3.1),

\|\bar{\partial}\gamma_{l}\|^{2}_{m\varphi_{U},h,\omega_{l}}=(\gamma_{l},\bar{\partial}^{\ast}_{m\varphi_{U},h}\bar{\partial}\gamma_{l})_{m\varphi_{U},h,\omega_{l}}+[\gamma_{l},e(\bar{\partial}f^{\ast}r_{U})^{\ast}\bar{\partial}\gamma_{l}]_{m\varphi_{U},h,\omega_{l}}=0.

We conclude that $\bar{\partial}\gamma_{l}=0$ . Equivalently, $\beta_{l}=s\alpha_{l}$ on $f^{-1}(U)\cap X^{o}$ . Now

\begin{split}\lim\|(s\alpha)|_{f^{-1}(U)\cap X^{o}}-\beta_{l}\|^{2}_{m\varphi_{U},h,\omega}&=\lim\|(s\alpha)|_{f^{-1}(U)\cap X^{o}}-s\alpha_{l}\|^{2}_{m\varphi_{U},h,\omega}\\ &\leqslant\sup_{X}(|s|^{2}e^{-(m-1)\varphi_{U}})\lim\|\alpha|_{f^{-1}(U)\cap X^{o}}-\alpha_{l}\|^{2}_{\varphi_{U},h,\omega}\\ &=0.\end{split}

In summary,

s\alpha\in\mathcal{H}^{n,q}(f^{-1}(U),L^{m}\otimes E(h)\otimes\mathscr{I}(m\varphi),f^{\ast}r_{U}).

Then we have successfully proved that

\otimes s:\mathcal{H}^{n,q}(f^{-1}(U),L\otimes E(h)\otimes\mathscr{I}(\varphi_{U}),f^{\ast}r_{U})\rightarrow\mathcal{H}^{n,q}(f^{-1}(U),L^{m}\otimes E(h)\otimes\mathscr{I}(m\varphi_{U}),f^{\ast}r_{U})

is well-defined. It is similar for a general Stein subset $V\subseteq U$ . As a result, the sheaf morphism

\otimes s:f_{\ast}\mathcal{H}^{n,q}(L\otimes E(h)\otimes\mathscr{I}(\varphi))\rightarrow f_{\ast}\mathcal{H}^{n,q}(L^{m}\otimes E(h)\otimes\mathscr{I}(m\varphi))

is also well-defined. The injectivity is then obvious.

Observe that if we substitute $s$ by an arbitrary holomorphic function $g$ on $X$ , everything is still going smoothly. Thus we have the following conclusion: the map induced by multiplied with $g$ :

\times g:R^{q}f_{\ast}(K_{X}\otimes L\otimes E(h)\otimes\mathscr{I}(f,\|L\|))\rightarrow R^{q}f_{\ast}(K_{X}\otimes L\otimes E(h)\otimes\mathscr{I}(f,\|L\|))

is well-defined and injective. This immediately implies that $R^{q}f_{\ast}(K_{X}\otimes L\otimes E(h)\otimes\mathscr{I}(f,\|L\|))$ is torsion-free. ∎

4.2 A relative Nadel-type vanishing theorem

Firstly, let’s recall an absolute version of the Nadel-type vanishing theorem, which is a simple variant of [Wu22], Theorem 1.3.

Theorem 4.2.

Let $X$ be a compact Kähler manifold of dimension $n$ , and let $L$ be a pseudo-effective line bundle on $X$ . Let $X^{o}$ be a Zariski open subset of $X$ , and let $(E,h)$ be a tame, Nakano semi-positive vector bundle of rank $r$ over $X^{o}$ . Assume that there exists a section $s$ of some multiple $L^{m}$ such that $\{s=0\}\subseteq X^{o}$ . Then

H^{q}(X,K_{X}\otimes L\otimes E(h)\otimes\mathscr{I}(\|L\|))=0

for $q>n-\kappa(L)$ .

(Sketch of Proof).

The proof is almost the same as [Wu22] except minor adjustment, so we only outline the main stream here.

We first use a log-resolution $\mu:\tilde{X}\rightarrow X$ to reduce the vanishing as

H^{q}(\tilde{X},K_{\tilde{X}}\otimes\hat{L}\otimes\mu^{\ast}E(h)),

where $\hat{L}=\mu^{\ast}L\otimes\mathcal{O}_{\tilde{X}}(-\sum\lfloor\frac{\lambda_{i}}{p}\rfloor E_{i})$ . Note $E_{i}$ is the prime component of the exceptional divisor with certain coefficients $\frac{\lambda_{i}}{p}$ . Let $e_{i}$ be the defining section of $E_{i}$ .

Denote $\tilde{L}=\mu^{\ast}L\otimes\mathcal{O}_{\tilde{X}}(-\sum\frac{\lambda_{i}}{p}E_{i})$ . The crucial observation is that we can endow $\tilde{L}^{p}$ , which is a $\mathbb{Z}$ -bundle, a smooth metric $\psi$ with semi-positive associated curvature, and $\kappa(\tilde{L}^{p})=\kappa(L)$ . Now we solve the complex Monge–Ampère equations

i\Theta_{\tilde{L}^{p},\varphi_{\varepsilon}}+\varepsilon\omega>0\quad\textrm{and}\quad(i\Theta_{\tilde{L}^{p},\varphi_{\varepsilon}}+\varepsilon\omega)^{n}=C_{\varepsilon}\omega^{n}

for every $\varepsilon>0$ , to obtain smooth metrics $\{\varphi_{\varepsilon}\}$ on $\tilde{L}^{p}$ .

We endow $\hat{L}$ with singular metric

\phi_{\varepsilon}=\frac{1}{p}(\delta\varphi_{\varepsilon}+(1-\delta)\psi)+\sum\{\frac{\lambda_{i}}{p}\}\log|e_{i}|^{2},

where $\delta>0$ is a sufficiently small number which will be fixed later. Denote by $0<a_{1}\leqslant\cdots\leqslant a_{n}$ and $0<\hat{a}_{1}\leqslant\cdots\leqslant\hat{a}_{n}$ , respectively, the eigenvalues of the curvature forms $i\Theta_{\tilde{L}^{p},\varphi_{\varepsilon}}+\varepsilon\omega$ and $i\Theta_{\hat{L},\phi_{\varepsilon}}+\frac{2\varepsilon}{p}\omega$ at every point $x\in\tilde{X}$ , with respect to the base Kähler metric $\omega(x)$ . We apply (3.2) on $X^{o}$ for every $\hat{L}\otimes\mu^{\ast}E$ -valued $(n,q)$ -form $\alpha$ to obtain

\begin{split}\|\bar{\partial}\alpha\|^{2}_{\phi_{\varepsilon},h}+\|\bar{\partial}^{\ast}\alpha\|^{2}_{\phi_{\varepsilon},h}&=\|\partial^{\ast}_{\phi_{\varepsilon,h}}\alpha\|^{2}_{\phi_{\varepsilon},h}+(i\Theta_{\hat{L}\otimes\mu^{\ast}E,\phi_{\varepsilon},h}\Lambda\alpha,\alpha)_{\phi_{\varepsilon},h}\\ &\geqslant\int_{\tilde{X}}r(\hat{a}_{1}+\cdots+\hat{a}_{q}-\frac{2q\varepsilon}{p})|\alpha|^{2}_{\phi_{\varepsilon},h}dV_{\omega}.\end{split}

(4.1)

Return to the proof of vanishing. Let us take a cohomology class

[\beta]\in H^{q}(\tilde{X},K_{\tilde{X}}\otimes\hat{L}\otimes\mu^{\ast}E(h)).

By using the de Rham–Weil isomorphism, we obtain a representative $\beta$ which is a smooth $\hat{L}\otimes\mu^{\ast}E$ -valued $(n,q)$ -form.

Then we use the canonical $L^{2}$ -estimate [Dem12] against $\phi_{\varepsilon},h$ to obtain elements $u_{\varepsilon},v_{\varepsilon}$ such that $\beta=u_{\varepsilon}+\bar{\partial}v_{\varepsilon}$ . Moreover, (4.1) implies that

\|v_{\varepsilon}\|^{2}_{\phi_{\varepsilon},h}\leqslant\frac{2q\varepsilon}{p}\int_{\tilde{X}}\frac{1}{\hat{a}_{1}+\cdots+\hat{a}_{q}}|\beta|^{2}_{\phi_{\varepsilon},h}dV_{\omega}.

(4.2)

Denote $\gamma_{\varepsilon}:=\frac{q\varepsilon}{p(\hat{a}_{1}+\cdots+\hat{a}_{q})}$ , then

\gamma_{\varepsilon}\leqslant\min(1,C\delta^{-1}\varepsilon^{1-(n-\kappa(L))/q}(a_{q+1}\cdots a_{n})^{1/q})

for a universal constant $C$ . Here we use the fact that

C_{\varepsilon}=\frac{\int_{\tilde{X}}(c_{1}(\tilde{L}^{p})+\varepsilon\omega)^{n}}{\int_{\tilde{X}}\omega^{n}}\geqslant C^{\prime}\varepsilon^{n-\kappa(L)}

for a universal constant $C^{\prime}$ . Notice that

\int_{\tilde{X}}a_{q+1}\cdots a_{n}dV_{\omega}\leqslant\int_{\tilde{X}}(i\Theta_{\tilde{L}^{p},\varphi_{\varepsilon}}+\varepsilon\omega)^{n-q}\wedge\omega^{q}=(c_{1}(\tilde{L}^{p})+\varepsilon[\omega])^{n-q}[\omega]^{q}\leqslant C^{\prime\prime},

hence the functions $(a_{q+1}\cdots a_{n})^{1/q}$ are uniformly bounded in $L^{1}$ -norm as $\varepsilon$ tens to zero. So $\gamma_{\varepsilon}$ converges almost everywhere to zero as $\varepsilon$ tends to zero when $q>n-\kappa(L)$ .

In the end, some standard analysis shows that for a small enough $\delta$ (independent of $\varepsilon$ ),

\lim_{\varepsilon\rightarrow 0}\frac{2q\varepsilon}{p}\int_{\tilde{X}}\frac{1}{\hat{a}_{1}+\cdots+\hat{a}_{q}}|\beta|^{2}_{\phi_{\varepsilon},h}dV_{\omega}=\lim_{\varepsilon\rightarrow 0}2\int_{\tilde{X}}\gamma_{\varepsilon}|\beta|^{2}_{\phi_{\varepsilon},h}dV_{\omega}=0.

Therefore $v_{\varepsilon}$ converges to zero as $\varepsilon$ tends to zero. Equivalently, $\beta$ is actually a boundary, and the desired vanishing is proved. ∎

Now we should prove Theorem 1.2, i.e. a relative Nadel-type vanishing theorem.

Theorem 4.3 ((=Theorem 1.2)).

Let $L$ be a holomorphic line bundle with $\kappa(L,f)\geqslant 0$ . Assume that there exists a section $s$ of some multiple $L^{m}$ such that $\{s=0\}\subseteq X^{o}$ . Then

R^{q}f_{\ast}(K_{X}\otimes L\otimes E(h)\otimes\mathscr{I}(f,\|L\|))=0

for $q>l-\kappa(L,f)$ .

Proof.

Let $Z$ be the set of the critical value of $f$ , which is a closed subvariety of $Y$ . As a result, $X_{y}$ is a compact Kähler manifold when $y\in f(X^{o})\setminus Z$ . Then we apply Theorem 4.2 to obtain that

H^{q}(X_{y},K_{X_{y}}\otimes L|_{X_{y}}\otimes E(h)|_{X_{y}}\otimes\mathscr{I}(\|L|_{X_{y}}\|))=0

for $q>l-\kappa(L|_{X_{y}})$ .

Now we claim that there exists a subset $V$ of $Y$ with zero Lebesgue measure, such that $\kappa(L|_{X_{y}})=\kappa(L,f)$ and

\mathscr{I}(\|L|_{X_{y}}\|)=\mathscr{I}(f,\|L\|)

when $y\in Y\setminus V$ . In fact, since $f_{\ast}(L^{m})$ is torsion-free for all $m$ , there exists a closed subvariety $Z_{m}$ such that $f_{\ast}(L^{m})$ is locally free on $Y\setminus Z_{m}$ . Let $V=(\cup^{\infty}_{m=1}Z_{m})\cup Z$ , which has zero Lebesgue measure. Then every section in $H^{0}(X_{y},L^{m})$ extends to the whole $X$ and $\kappa(L|_{X_{y}})=\kappa(L,f)$ when $y\in Y\setminus V$ . Hence $\mathscr{I}(\|L|_{X_{y}}\|)=\mathscr{I}(f,\|L\|)$ by definition.

Now on $f(X^{o})\setminus V$ we obtain

R^{q}f_{\ast}(K_{X}\otimes L\otimes E(h)\otimes\mathscr{I}(f,\|L\|))=0

for $q>l-\kappa(L,f)$ . We then conclude that this vanishing actually holds on the whole $Y$ since $R^{q}f_{\ast}(K_{X}\otimes L\otimes E(h)\otimes\mathscr{I}(f,\|L\|))$ is torsion-free by Theorem 4.1. ∎

5 Applications

In this section we should discuss the applications on harmonic bundles. Firstly, combining Theorem 1.2 and Proposition 2.2, we obtain that

Corollary 5.1 ((=Corollary 1.2)).

R^{q}f_{\ast}(K_{X}\otimes L\otimes E(h)\otimes\mathscr{I}(f,\|L\|))=0

for $q>l-\kappa(L,f)$ .

In particular, Saito’s $S$ -sheaf can be represented as such an $E(h)$ . The original construction of S-sheaves for a real variation of polarized Hodge structure

(\mathbb{V},\nabla,\mathcal{F}^{\cdot},S)

is based on the theory of Hodge module. We recommend [Sai91] as the reference. However, the original definition of S-sheaves is not involved in our paper. Instead, let’s recall the following equivalent description provided in [ScY20].

Proposition 5.1 ((c.f. [ScY20], Theorem A)).

Let $(\mathbb{V},\nabla,\mathcal{F}^{\cdot},S)$ be a real variation of polarized Hodge structure on $X^{o}$ . Denote by $S(\mathbb{V})=\mathcal{F}^{p_{\max}}$ where $p_{\max}=\max\{p;\mathcal{F}^{p}\neq 0\}$ . Denote by $h$ the Hermitian metric on $S(\mathbb{V})$ induced by the polarization $S$ . Denote by $\mathrm{IC}_{X}(\mathbb{V})$ the intermediate extension of $\mathbb{V}$ on $X$ as a pure Hodge module and by $S(\mathrm{IC}_{X}(\mathbb{V}))$ the Saito’s S-sheaf associated to $\mathrm{IC}_{X}(\mathbb{V})$ . Then

K_{X}\otimes S(\mathbb{V})(h)\simeq S(\mathrm{IC}_{X}(\mathbb{V})).

Based on this result, we obtain that

Corollary 5.2 ((=Corollary 1.3)).

Let $L$ be a holomorphic line bundle on $X$ with $\kappa(L,f)\geqslant 0$ . Assume that there exists a section $s$ of some multiple $L^{m}$ such that $\{s=0\}\subseteq X^{o}$ . Then

R^{q}f_{\ast}(S(\mathrm{IC}_{X}(\mathbb{V}))\otimes L\otimes\mathscr{I}(f,\|L\|))=0

for $q>l-\kappa(L,f)$ .

Proof.

Let $(H,\theta,h_{H})$ be the tame harmonic bundle associated with $(\mathbb{V},\nabla,\mathcal{F}^{\cdot},S)$ . In order to apply Theorem 1.2, in the view of Proposition 2.3 and Proposition 5.1, it is enough to show that $(S(\mathbb{V}),h)$ is Nakano semi-positive. There are two methods available.

In the view of Proposition 2.2, we could show that $\bar{\theta}(S(\mathbb{V}))=0$ . However, it is obvious for the reason of degree. Alternatively, the Nakano semi-positivity is the direct consequence of Schmid’s curvature calculation. (See [Sch73], Lemma 7.18.) ∎

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